The functor \(\sigma \rightarrow S_\sigma \)  gives an equivalence of categories between rational polyhedral cones and the category of sharp, fine and saturated monoids. We can thus also think of \(\mathcal {J}^{trop}_{g,n}\)  (and similarly of \(\mathcal {J}^{trop}_{g,n,d}\) , \(\mathcal {J}^{trop, spl}_{g,n, (d)}\)  and \(\mathcal {J}^{trop}_{g,n}(\phi )\) ) as a category fibered in groupoids over the category of sharp fine and saturated monoids. We may expand \(\mathcal {J}^{trop}_{g,n}\)  to a category fibered in groupoids over the category \(\textbf {ShpMon}^{op}\)  of sharp integral and saturated monoids, whose fiber over a sharp integral and saturated monoid \(P\)  is the groupoid of pairs \((\Gamma ,D)\)  consisting of a quasistable tropical curve \(\Gamma \)  of genus \(g\)  with \(n\)  marked legs, whose edges are metrized by the monoid \(P\) , and an admissible divisor \(D\)  on \(\mathbb {G}(\Gamma )\) . We refer the interested reader to [1]} for more background on this procedure.
